I've never heard of parameterised tests being called property based testing - they don't define a property just a list of cases. At that point we may as well call all unit tests property based tests and lose any useful distinction.
I agree that calling table based testing 'property based testing' is stretching the definition.
> Nothing in property testing says who has to choose the values used.
I take issue with that. Properties of interest are some variant of 'for all x from some set s, some property holds.'
Then instead of testing all possible x, we randomly sample. But it is important that our tests define the set s.
When we do example based testing (whether traditional unit testing, or table based testing), the set s is usually only implied. So you might be testing your functionality for strings 'foo', and 'bar' and perhaps 'foo bar'. But it's not clear, whether you also mean it to pass for the string '' or '안녕하세요' or '\t\n'. That's at best implied, or perhaps vaguely described in a human readable comment.
